ふっふ、ほっほ 下記のAI による概要で ”Theorem: Let X and Y be metric spaces, S a subset of X, and f: S -> Y. If f is uniformly continuous and Y is complete, then there exists a unique continuous extension of f to ¯S (the closure of S). Furthermore, this extension is uniformly continuous.”と言ってますよ ”有界閉区間”の条件はありません！！ｗ ；ｐ） ＜キーワード＞ 数学 距離空間 稠密 関数 一様連続 拡張 ↓英訳 Mathematics Metric space Dense Function Uniform continuity Extension ↓検索 googleさん AI による概要（AI responses may include mistakes. Learn more） In the context of metric spaces, if a function f is uniformly continuous on a dense subset S of a complete metric space X, then f can be extended to a uniformly continuous function F defined on the entire space X. This theorem is a powerful tool for extending functions from dense subsets to the whole space while preserving uniform continuity, which is crucial in many mathematical applications.
Key Concepts and Definitions: Metric Space: A set equipped with a distance function (or metric) that satisfies certain properties. Dense Subset: A subset where every point in the larger space is either in the subset or can be approached arbitrarily closely by a point in the subset. Uniformly Continuous Function: A function where the distance between the function values of two points can be made arbitrarily small as long as the distance between the two points is small, regardless of where those points are in the domain.
Complete Metric Space: A metric space where every Cauchy sequence (a sequence that gets arbitrarily close to each other) converges to a point in the space. Theorem: Let X and Y be metric spaces, S a subset of X, and f: S -> Y. If f is uniformly continuous and Y is complete, then there exists a unique continuous extension of f to ¯S (the closure of S). Furthermore, this extension is uniformly continuous.
